Three-dimensional stress and strain distribution in a two-layer model of a coronary artery

For a right coronary artery, three-dimensional stress and strain distributions at a physiological intraluminal pressure and an axial extension ratio were computed on the basis of a two-layer elastic model. To validate the model, curves of external radius versus pressure and of axial force versus pressure were computed for three axial extension ratios. To analyze mechanical properties, stress-free configurations of media and adventitia, and the constitutive law of each layer in literature, were used. The present study showed that the peak circumferential stress and the peak axial stress appear in the media at the boundary between the media and adventitia. This result is due to the opening angle of the media being larger than π (rad) and the larger value of a material constant of the strain energy function for the media than for the adventitia. The circumferential stress and strain were discontinuous at the boundary. On the other hand, the radial stress was continuous at the boundary because of the boundary condition for stress. The circumferential stress and axial stress in the adventitia were almost uniformly distributed, and smaller than in the media. The residual stress and strain were also computed. The circumferential residual stress and strain were almost linearly distributed in each layer, although discontinuity appeared at the boundary between the two layers.     

Transmission characteristics of axial waves in blood vessels

The elastic behavior of blood vessels can be quantitatively examined by measuring the propagation characteristics of waves transmitted by them. In addition, specific information regarding the viscoelastic properties of the vessel wall can be deduced by comparing the observed wave transmission data with theoretical predictions. The relevance of these deductions is directly dependent on the validity of the mathematical model for the mechanical behavior of blood vessels used in the theoretical analysis. Previous experimental investigations of waves in blood vessels have been restricted to pressure waves even though theoretical studies predict three types of waves with distinctly different transmission characteristics. These waves can be distinguished by the dominant displacement component of the vessel wall and are accordingly referred to as radial, axial and circumferential waves. The radial waves are also referred to as pressure waves since they exhibit pronounced pressure fluctuations. For a thorough evaluation of the mathematical models used in the analysis it is necessary to measure also the dispersion and attenuation of the axial and circumferential (torsion) waves.

To this end a method has been developed to determine the phase velocities and damping of sinusoidal axial waves in the carotid artery of anesthetized dogs with the aid of an electro-optical tracking system. For frequencies between 25 and 150 Hz the speed of the axial waves was between 20 and 40 m/sec and generally increased with frequency, while the natural pressure wave travelled at a speed of about 10 m/sec. On the basis of an isotropic wall model the axial wave speed should however be approximately 5 times higher than the pressure wave speed. This discrepancy can be interpreted as an indication for an anisotropic behavior of the carotid wall. The carotid artery appears to be more elastic in the axial than in the circumferential direction.     

The highly elongated Drosophila mechanosensory bristle - a new model for studying polarized microtubule function

Microtubules (MTs) are polar polymers that can facilitate asymmetric distribution of cell components, a process important for polarized cell growth. The highly elongated and polarized Drosophila mechnosensory bristle cytoplasm is filled with short MTs that constitute a significant component of the shaft cytoplasm. Inhibition of MT assembly affects biased axial growth in the bristle and highlights the importance of MTs for this process. We demonstrate that the vast majority of bristle MTs are organized in a polarized manner, minus-ends out. We also show that genetic disruption of the MT polarity affects the polar distribution of cell components and leads to an alteration in the biased axial shape of the bristle shaft. Thus, we suggest that the asymmetric organization of the MT population within the bristle cell shaft is necessary for the proper axial elongation of this cellular extension. We would also like to emphasize the benefits of using the bristle cell as a model for studying MTs and MT-binding proteins because changes to this cytoskeletal component result in easily recognized at the phenotypes.     

Measurement of particle sizes by elastic and quasi-elastic light scattering.

A method of measuring an average particle radius in a highly polydisperse dispersion using the wavelength dependence of turbidity is described. For particles which are no larger than 0.3 of the wavelength of light used, a polynomial representation of the scattering cross-section can be used. For larger particles, more extensive numerical calculations are required. The use of the method is illustrated by determining the average particle radius of casein micelles by both elastic and quasi-elastic light scattering techniques. A polydisperse homogeneous sphere model is found to be a reasonably accurate representation of casein micelles. Several modifications of the model which would improve the agreement between the two techniques are mentioned briefly.     

Effects of the cross-linkers on the buckling of microtubules in cells

In cells, the protein cross-linkers lead to a distinct buckling behavior of microtubules (MTs) different from the buckling of individual MTs. This paper thus aims to examine this issue via the molecular structural mechanics (MSM) simulations. The transition of buckling responses was captured as the two-dimensional-linkers were replaced by the three-dimensional (3D) ones. Then, the effects of the radial orientation and the axial density of the 3D-linkers were examined, showing that more uniform distribution of the radial orientation leads to the higher critical load with 3D buckling modes, while the inhomogeneity of the axial density results in the localized buckling patterns. The results demonstrated the important role of the cross-linker in regulating MT stiffness, revealed the physics of the experimentally observed localized buckling and these results will pave the way to a new multi-component mechanics model for whole cells.     

Microtubule length and dynamics: Boundary effect and properties of extended radial array

In interphase cells, microtubules (MT) form an extended radial array. The length of individual MTs in living cells exhibits substantial stochastic fluctuations, while the average length distribution in a cell remains nearly constant. We present a quantitative model that describes the relation of the MT length and dynamics in the steady state in the cell using the minimal set of parameters (cell radius, tubulin concentration, critical concentration for plus-end elongation and the number of nucleation sites). The MT array is approximated as a radial system, where minus-ends of MTs are associated with nucleation sites on the centrosome, while plus ends grow and shorten. Dynamic instability of MT plus ends is approximated as a random walk process with boundary conditions; the behavior of an MT array is quantified using diffusion and drift coefficients (Vorobjev et al., 1997; Vorobjev et al., 1999). We show that the establishment of the extended steady-state array could be accomplished solely by the limitation of MT growth by the cell margin. For the cell radius, tubulin concentration, critical concentration for plus-end elongation, and the number of nucleation sites we determined the reference point in the parameter space where plus ends of individual MTs, on average, neither elongate nor shorten. In this case, the average MT length is equal to the half of the cell radius. When any parameter is shifted from its reference value, MTs become longer or shorter and, consequently, acquire a positive or negative drift of their plus ends. In the vicinity of the reference point, a change in any parameter has a major effect on the MT length and a rather small effect on the drift. When the average MT length is close to the cell radius, the drift of free plus ends becomes substantial, resulting in processive growth of individual MTs in the internal cytoplasm, accompanied by the apparent stabilization of plus ends at the cell margin. Under these conditions small changes in parameters have a significant impact on the magnitude of the drift. Experimental analysis of MT plus-end dynamics in different cultured cells shows that, in most cases, plus ends display positive drift, which, in the framework of the presented model, is in agreement with the simultaneous presence of long MTs.     

Dynamic visco-elastic buckling analysis of an in vitro airway model: prediction of circumferential and axial buckling mode numbers

The purpose of the research reported here was to elucidate the mechanism of formation of the various lobes observed in asthmatic airways by both theoretical and experimental analysis employing an in vitro airway model. The rationale is that the elucidated mechanism will facilitate the development of new diagnostic methods and treatment regimens for asthma. Lobe formation was analyzed on the basis of an assumption of cross-sectional buckling of the airway. Here, we propose a dynamic visco-elastic buckling model analysis of the airway for the prediction of circumferential and axial buckling mode numbers. The calculated circumferential buckling mode numbers were in reasonably good agreement with those measured in the dynamic buckling experiment using the in vitro airway model. The calculated axial buckling mode numbers were in qualitative agreement with those observed in the experiment. The non-dimensional parameters related to the remodeling and the consequent pathologies occurring in asthmatic airways were also shown, and the influence of changes in the non-dimensional parameters on the circumferential and axial buckling mode numbers was also calculated. The circumferential and axial buckling mode numbers decreased due to thickening and stiffening of the basement membrane. Thickening of the tissues surrounding the basement membrane and the increase in the mucus secreted in the airway lumen were modeled as an increase in the added mass on the basement membrane. The results of calculation showed that the circumferential and axial buckling mode numbers increased because of the thickening of the surrounding tissues and the increase in mucus secretion. We suggest that it may be possible to diagnose the severity of asthma by using the results of the calculation of the changes in the buckling mode numbers caused by the changes in the strength of the remodeling. The physiological reality of the in vitro airway model reported here is discussed using the non-dimensional parameters.     

A model for passive elastic properties of rat vena cava

A two-dimensional model for the elastic properties of vena cava abdominalis under orthotropic deformation is introduced and tested against the experimental data obtained from six specimen of rat venae cavae by pressurization experiments. The model is based on membrane approximation and suited for vessels where most of the elastic elements are oriented axially, while circumferential contraction is exerted by redirecting axial stress by some network of oblique fibers. For the experimental data considered in this paper, the ratio between axial and circumferential stress depends almost exclusively on the circumferential extension ratio. As a consequence, the mechanical system can be formally decomposed in a kinematic system reacting by axial contraction on circumferential extension without any loss or storage of energy, serially connected to a hyperelastic system acting only in axial direction. Both systems are modeled separately by equations obtained by a purely phenomenological approach with two parameters for each system. This leads to reasonable reproduction of the experimental data. Introducing a correction parameter, which takes into account that the model assumption on the decomposition does not hold exactly, we get better reproduction of data. However, this is paid for by loss of physical rigor and in particular by departing from the assumption of hyperelasticity.     

Influence of reactor geometry on mixing characteristics in a down flow jet loop bioreactor: newtonian and non-newtonian fluids

Mixing characteristics in the downcomer and the riser of a continuous down-flow jet loop bioreactor was studied with Newtonian and non-Newtonian fluids. The mixing parameters were determined through the curve fitting of the experimental impulse response data with the solution of one dimensional axial dispersion model. It was found that circulation number and axial dispersion coefficient increased with an increase in liquid flow rate and draft tube to column diameter ratio and the axial dispersion coefficient was comparatively higher in the riser. The circulation number increased with decrease in nozzle diameter. The model predicted the experimental data well within 8% deviation for both the systems (water and CMC). Correlations were obtained to predict axial dispersion coefficients in the riser and downcomer of the reactor.     

Bolus contaminant dispersion in oscillating flow in curved tubes.

The investigation of longitudinal dispersion of tracer substances in unsteady flows has biomechanical application in the study of heat and mass transport within the bronchial airways during normal, abnormal, and artificial pulmonary ventilation. To model the effects of airway curvature on intrapulmonary gas transport, we have measured local gas dispersion in axially uniform helical tubes of slight pitch during volume-cycled oscillatory flow. Following a small argon bolus injection into the flow field, the time-averaged effective diffusion coefficient (Deff/Dmol) for axial transport of the contaminant was evaluated from the time-dependent local argon concentration measured with a mass spectrometer. The value of (Deff/Dmol) is extracted from the curve of concentration versus time by two techniques yielding identical results. Experiments were conducted in two helical coiled tubes (delta = 0.031, lambda = 0.022 or delta = 0.085, lambda = 0.060) over a range of 2 < alpha < 15, 3 < A < 15, where delta is the ratio of tube radius to radius of curvature, lambda is the ratio of pitch height to radius of curvature, alpha is the Womersley parameter or dimensionless frequency, and A is the stroke amplitude or dimensionless tidal volume. Experimental results show that, when compared to transport in straight tubes, the effective diffusivity markedly increases in the presence of axial curvature. Results also compare favorably to mathematical predictions of bolus dispersion in a curved tube over the ranges of frequency and tidal volume studied.     

Structure–property relation and relevance of beam theories for microtubules: a coupled molecular and continuum mechanics study

Quasi-one-dimensional microtubules (MTs) in cells enjoy high axial rigidity but large transverse flexibility due to the inter-protofilament (PF) sliding. This study aims to explore the structure–property relation for MTs and examine the relevance of the beam theories to their unique features. A molecular structural mechanics (MSM) model was used to identify the origin of the inter-PF sliding and its role in bending and vibration of MTs. The beam models were then fitted to the MSM to reveal how they cope with the distinct mechanical responses induced by the inter-PF sliding. Clear evidence showed that the inter-PF sliding is due to the soft inter-PF bonds and leads to the length-dependent bending stiffness. The Euler beam theory is found to adequately describe MT deformation when the inter-PF sliding is largely prohibited. Nevertheless, neither shear deformation nor the nonlocal effect considered in the ‘more accurate’ beam theories can fully capture the effect of the inter-PF sliding. This reflects the distinct deformation mechanisms between an MT and its equivalent continuous body.     

On the relation of the microtubule length and dynamics: boundary effect and properties of an extended radial array

In interphase cells, microtubules (MT) are long and form extended radial array. The length of individual MTs in living cells exhibits substantial stochastic fluctuations while the average length distribution in a cell remains nearly constant. We present a quantitative model that describes relation of the MT length and dynamics in the steady state in the cell using the minimal set of parameters (cell radius, tubulin concentration, critical concentration for plus end elongation, and the number of nucleation sites). The MT array is approximated as a radial system, where MT minus ends are associated with the nucleation sites on the centrosome, while plus ends grow and shorten. Dynamic instability of MT plus ends is approximated as a random walk process with boundary conditions and the behavior of MT array is quantified using diffusion and drift coefficients (Vorobjev et al., 1997, 1999). We show that establishment of the extended steady-state array could be accomplished solely by the limitation of the MT growth by the cell margin. We determined for the cell radius, tubulin concentration, critical concentration for plus end elongation, and number of nucleation sites the reference point in the parameter space where plus ends of individual MT on average neither elongate nor shorten. In this case average length of MT is equal to the half of cell radius. When any parameter is shifted from its reference value MTs become longer or shorter and consequently acquire positive or negative drift of their ends. In the vicinity of reference point, change in any parameter has major effect on the MT length and rather small effect on the drift. When mean length of the MTs is close to the cell radius the drift of the free plus ends becomes substantial, resulting in processive growth of individual MTs in the internal cytoplasm accompanied by apparent stabilization of the plus ends at the cell margin. Under these conditions small changes in parameters have significant impact on the magnitude of drift. Experimental analysis of the MT plus ends dynamics in different cultured cells shows that in most cases plus ends display positive drift, which, in the framework of the presented model, is in agreement with the simultaneous presence of long MTs.     

Slide-and-cluster models for spindle assembly

BACKGROUND: Mitotic and meiotic spindles are assemblies of microtubules (MTs) that form during cell division to physically separate sister chromosomes. How the various components of spindles act together to establish and maintain the dynamic bipolar structure of spindles is not understood. Interactions between MTs and motors have been studied both experimentally and theoretically in many contexts, including the self-organization of arrays of MTs by motors and the competition between different classes of motors to move a single load. This work demonstrates how the interplay between two types of motors together with continual nucleation of MTs by chromosomes could organize the MTs into spindles. RESULTS: We propose a slide-and-cluster model based on four known molecular activities: MT nucleation near chromosomes, the sliding of MTs by a plus-end-directed motor, the clustering of their minus ends by a minus-end-directed motor, and the loss of MTs by dynamic instability. Our model applies to overlapping, nonkinetochore MTs in anastral spindles, and perhaps also to interpolar MTs in astral spindles. We show mathematically that the slide-and-cluster mechanism robustly forms bipolar spindles with sharp poles and a stable steady-state length. This model accounts for several experimental observations that were difficult to explain with existing models. Three new predictions of the model were tested and verified in Xenopus egg extracts. CONCLUSIONS: We show that a simple two-motor model could create stable, bipolar spindles under a wide range of physical parameters. Our model is the first self-contained model for anastral spindle assembly and MT sliding (known as poleward flux). Our experimental results support the slide-and-cluster scenario; most significantly, we find that MT sliding slows near spindle poles, confirming the model's primary prediction.     

Dynamics and the life cycle of cell microtubules

In living cells microtubules (MTs) continuously grow and shorten. This feature of MTs was discovered in vitro and named dynamic instability. Comparison of dynamic instability of MTs in vitro and in vivo shows a number of differences. MTs in vivo rapidly grow (up to 20 microns/min), duration of their shortening is small (on average 15-20 s), and pauses are prominent. In different animal cells MTs grow from the centrosome and form a radial array. In such cells growth of MTs is persistent, i.e. undergo without interruptions until plus end of a MT reaches cell margin. Analysis of literature and original data shows that interconvertion between phases of growth, shortening and pause is asymmetric: growth often converts into pause, while shortening always converts into growth without pause. We suggest dynamic instability described near the cell margin in numerous publications results not only from intrinsic properties of MTs, but also because of the external obstacles for their growth. MT behavior in the cells with radial array of long MTs could be treated as dynamic instability with boundary conditions. One boundary is the centrosome responsible for rapid initiation of MT growth. Another boundary is cell margin limiting MT elongation. MT growth occurs with constant mean velocity, and potential duration of growth phase might exceed cell radius. MT shortening is usually smaller than MT length however velocity of shortening increases with time. Random episodes of rapid shortening are sufficient for the exchange of MTs in 10-20 min in the cells not more than 40-50 microns in diameter. Experimental data show that similar rate of exchange of MTs is in the large cells. This is achieved employing another mechanism, namely release of MTs and depolymerization from the minus end. In the minus end pathway time required for the exchange of MTs does not depend on cell radius and is determined primarily by the frequency of releases. Thus a small number of free MTs with metastable minus ends significantly reduce time required for the renovation of the radial MT array. Summarizing all experimental data we suggest the life cycle scheme for the MT in a cell. MT is initiated at the centrosome and grows rapidly until it reaches cell margin. At the margin the plus end oscillates, and finally MT depolimerizes. MT "death" comes from a random catastrophe (shortening from the plus end) in small cells or from release and depolymerization of the minus end in large cells.     

A Highlights from MBoC Selection: Regulation of microtubule-based transport by MAP4

Microtubule (MT)-based transport of organelles driven by the opposing MT motors kinesins and dynein is tightly regulated in cells, but the underlying molecular mechanisms remain largely unknown. Here we tested the regulation of MT transport by the ubiquitous protein MAP4 using Xenopus melanophores as an experimental system. In these cells, pigment granules (melanosomes) move along MTs to the cell center (aggregation) or to the periphery (dispersion) by means of cytoplasmic dynein and kinesin-2, respectively. We found that aggregation signals induced phosphorylation of threonine residues in the MT-binding domain of the Xenopus MAP4 (XMAP4), thus decreasing binding of this protein to MTs. Overexpression of XMAP4 inhibited pigment aggregation by shortening dynein-dependent MT runs of melanosomes, whereas removal of XMAP4 from MTs reduced the length of kinesin-2–dependent runs and suppressed pigment dispersion. We hypothesize that binding of XMAP4 to MTs negatively regulates dynein-dependent movement of melanosomes and positively regulates kinesin-2–based movement. Phosphorylation during pigment aggregation reduces binding of XMAP4 to MTs, thus increasing dynein-dependent and decreasing kinesin-2–dependent motility of melanosomes, which stimulates their accumulation in the cell center, whereas dephosphorylation of XMAP4 during dispersion has an opposite effect.     

Torque generation by one of the motor subunits of heterotrimeric kinesin-2

Heterotrimeric kinesin-2 motors [1] and [2] transport intraflagellar transport (IFT)-particles from the base to the tip of the axoneme to assemble and maintain cilia [3], [4], [5], [6], [7], [8], [9] and [10]. These motors are distinct in containing two non-identical motor subunits together with an accessory subunit [1], [11], [12], [13], [14] and [15]. We evaluated the significance of this organization by comparing purified wild type kinesin-2 holoenzymes that support IFT in vivo, with mutant trimers containing only one type of motor domain that do not support IFT in vivo. In motility assays, wild type kinesin-2 moved microtubules (MTs) at a rate intermediate between the rates supported by the two mutants. Interestingly, one of the mutants, but not the other mutant or the wild type protein, was observed to drive a persistent counter-clock-wise rotation of the gliding MTs. Thus one of the two motor domains of heterotrimeric kinesin-2 exerts torque as well as axial force as it moves along a MT, which may allow kinesin-2 to control its circumferential position around a MT doublet within the cilium.     

Nonlocal shear deformable shell model for postbuckling of axially compressed microtubules embedded in an elastic medium

Buckling and postbuckling analysis is presented for axially compressed microtubules (MTs) embedded in an elastic matrix of cytoplasm. The microtubule is modeled as a nonlocal shear deformable cylindrical shell which contains small scale effects. The surrounding elastic medium is modeled as a Pasternak foundation. The governing equations are based on higher order shear deformation shell theory with a von Kármán-Donnell-type of kinematic nonlinearity and include the extension-twist and flexural-twist couplings. The thermal effects are also included and the material properties are assumed to be temperature-dependent. The small scale parameter e ₀ a is estimated by matching the buckling load from their vibrational behavior of MTs with the numerical results obtained from the nonlocal shear deformable shell model. The numerical results show that buckling load and postbuckling behavior of MTs are very sensitive to the small scale parameter e ₀ a. The results reveal that the MTs under axial compressive loading condition have an unstable postbuckling path, and the lateral constraint has a significant effect on the postbuckling response of a microtubule when the foundation stiffness is sufficiently large.     

Spatial relationship between microtubules and plasma-membrane rosettes during the deposition of primary wall microfibrils in Closterium sp.

The mechanism by which cortical microtubules (MTs) control the orientation of cellulose microfibril deposition in elongating plant cells was investigated in cells of the green alga, Closterium sp., preserved by ultrarapid freezing. Cellulose microfibrils deposited during formation of the primary cell wall are oriented circumferentially, parallel to cortical MTs underlying the plasma membrane. Some of the microfibrils curve away from the prevailing circumferential orientation but then return to it. Freeze-fracture electron microscopy shows short rows of particle rosettes on the P-face of the plasma membrane, also oriented perpendicular to the long axis of the cell. Previous studies of algae and higher plants have provided evidence that such rosettes are involved in the deposition of cellulose microfibrils. The position of the rosettes relative to the underlying MTs was visualized by deep etching, which caused much of the plasma membrane to collapse. Membrane supported by the MTs and small areas around the rosettes resisted collapse. The rosettes were found between, or adjacent to, MTs, not directly on top of them. Rows of rosettes were often at a slight angle to the MTs. Some evidence of a periodic structure connecting the MTs to the plasma membrane was apparent in freeze-etch micrographs. We propose that rosettes are not actively or directly guided by MTs, but instead move within membrane channels delimited by cortical MTs attached to the plasma membrane, propelled by forces derived from the polymerization and crystallization of cellulose microfibrils. More widely spaced MTs presumably allow greater lateral freedom of movement of the rosette complexes and result in a more meandering pattern of deposition of the cellulose fibrils in the cell wall.Abbreviations E-face exoplasmic fracture face - MT microtubule - P-face protoplasmic fracture-face     

Flow-induced shear strain in intima of porcine coronary arteries.

The in vivo circumferential strain has a small variation throughout the vascular system (aorta to arterioles). The axial strain has also been shown to be nearly the same as the circumferential strain under physiological loading. Since the endothelium is mechanically much softer than the media-adventitia in healthy arteries, the porcine intima was considered as a mechanically distinct layer from the media-adventitia in a two-layer computational model. Based on the simulation result, we hypothesize that the flow-induced shear strain in intima can be of similar value as the pressure-induced circumferential strain in healthy coronary arteries, even though the shear stress is orders of magnitude smaller than the circumferential stress. The nearly isotropic deformation (circumferential, axial, and shear strains) may have important implications for mechanical homeostasis of endothelial cells, mechanotransduction, growth, and remodeling of blood vessels.     

Quantifying turbulent wall shear stress in a stenosed pipe using large eddy simulation

Large eddy simulation was applied for flow of Re=2000 in a stenosed pipe in order to undertake a thorough investigation of the wall shear stress (WSS) in turbulent flow. A decomposition of the WSS into time averaged and fluctuating components is proposed. It was concluded that a scale resolving technique is required to completely describe the WSS pattern in a subject specific vessel model, since the poststenotic region was dominated by large axial and circumferential fluctuations. Three poststenotic regions of different WSS characteristics were identified. The recirculation zone was subject to a time averaged WSS in the retrograde direction and large fluctuations. After reattachment there was an antegrade shear and smaller fluctuations than in the recirculation zone. At the reattachment the fluctuations were the largest, but no direction dominated over time. Due to symmetry the circumferential time average was always zero. Thus, in a blood vessel, the axial fluctuations would affect endothelial cells in a stretched state, whereas the circumferential fluctuations would act in a relaxed direction.